Events in fields of optical vortices: rings and reconnection

It is known (Berry and Dennis 2007 J. Phys. A: Math. Theor. 40 65–74; Berry and Dennis 2012 Eur. J. Phys. 33 723–731) that only one kind of reaction between wave vortices can occur generically in a monochromatic optical field. It appears either in elliptic form as the birth and death of vortex rings or in hyperbolic form as reconnection between separate vortex lines. To make it occur the field must be changed, and, since the codimension is one, it suffices to adjust a single external parameter. The paper analyses a model in which the initial field is produced by superposing n plane waves of the same frequency but different random amplitudes, directions and phases. This is perturbed by an additional plane wave of variable amplitude. The field necessarily obeys the Helmholtz equation and, in spite of the randomness, there is systematic behaviour for n = 3 and 4, which leads to some understanding of the more complicated results for higher values of n. Three plane waves of equal amplitude, perturbed by a fourth, provide a surprising special case, and the remarkable succession of events discovered by (O'Holleran et al 2006a J. Eur. Opt. Soc. Rapid Publ. 1 06008; O'Holleran et al 2006b Opt. Express 14 3039–3044) is fully explained. This is a central point of the paper. Looking at the singularity itself, and initially following Berry and Dennis, the simplest model that satisfies the Helmholtz equation is presented and also the most general local model that uses 'polynomial waves'. We also consider waves that are described simply by a polynomial without any exponential factor. The inclusion of time in the polynomial allows explicitly for quasi-monochromatic waves in which the events occur spontaneously, rather than by adjusting an external control. The circulating phase structure around a simple wave vortex is its most distinctive feature. But in reconnection two such singular vortex lines cross one another and the phase pattern around them must reflect this higher singularity. How it does so is illustrated in the paper.